# Why does multiplying a number on a clock face by 10 and then halving, give the minutes? ${}{}$

My daughter in grade 3 is learning about telling time at her school. She eagerly showed me this method she has discovered on her own to tell the minutes part of the time on an analogue clock. I wasn't sure at first because I have never heard about it before but it works really well.

Here is her method:

• Look at the minute hand and see what number it is pointing to, let's say it's $3$.

• Add a zero at the end to make it $30$.

• Halve it to get the minutes, so $3$ becomes $30$ and halving it gives $15$, $6$ becomes $60$ and halving it gives $30$ and so on.

It works well but I am not sure why. What mathematically justifies the method used?

• @Rafee This is a known defect in many clocks – pjs36 Mar 21 '18 at 2:48
• @JustinLardinois I imagine many people just learn it by rote - the same way many people learn their times tables. "6 7s are 42" is just a fact that they know to be true in isolation from any interpretation of how that works, and so is "the long hand pointing at the 4 is '20 past'". – Brondahl Mar 21 '18 at 9:24
• @MontyHarder, that's how you can tell a clock was made by a mathematician! – G Tony Jacobs Mar 21 '18 at 15:51
• "it works really well" ? You mean it work when the minute hand is exactly on an hour (demarker). I would not call that "really well", as it eliminates all other possible times. You also say "I am not sure why? " the equation you describe is (x * 10)/2 = 5X (as described below). Your profile indicates that you are a Business analyst(?) are you just trying to get up votes on your question? – DaniDev Mar 21 '18 at 20:36
• @FelixMarin She isn't 3 years old; she's in grade 3. – Your IDE Mar 22 '18 at 8:26

Each number on the clock face is worth five minutes. One good way to multiply by $5$ is to first multiply by $10$, and then divide by $2$. This works because $5=10\div 2$.

• You can also divide by $5$ in a similar fashion: Multiply by $2$ and then divide by $10$. – G Tony Jacobs Mar 21 '18 at 2:54
• You should encourage your daughter to come up with as many mathematical tricks like this as possible. As she gets older, she should be able to explain more clearly why it always works (or when a trick shouldn't be used). Not only will it give her more insight into math and be more efficient, but she might have more fun at the same time. As she gets better at multiplying (sounds like she is ready to start learning more), see if you can get her to reason out how to quickly multiply by 9 or 99, etc. – Nick Brown Mar 21 '18 at 13:30
• @NickBrown This is the sort of thing my dad taught me long ago. He worked as an auditor for A&P stores long before computers or even electronic calculators, and he took inventories often enough to give Count von Count a challenge for best counter. He learned/created a lot of tricks to be good at it. – Monty Harder Mar 21 '18 at 15:14
• You can also divide by $2$ in a similar fashion: Multiply by $216$ and then divide by $432$. :-) – Asaf Karagila Mar 21 '18 at 17:24
• @AsafKaragila , I always do that when working in base $432$ 🙂 – G Tony Jacobs Mar 21 '18 at 19:05

The minute hand passes over $12$ numbers in $60$ minutes.

That is $5$ minutes for each number.

Note that multiplying by $5$ is the same as multiplying by $10$ and dividing by $2$

Thus $3$ translates into $15$ which is $3(10)/2$.

Similarly $6$ translates into $30$ which is $6(10)/2$.

• +1 for explaining the significance of 12 and 60, which we need to explain how come each "number" represents 5 minutes rather than any other number. – Rosie F May 27 '18 at 6:13

The minutes are reckoned as

(Minutes pointing numeral N)(number of minutes in one hour =60)/(maximum minutes digits available on dial= 12) $= 5 N$

The procedure your daughter gave has effect of multiplying pointed figure $N$ by $5$; .. so it works.

From the perspective of the minute hand, the numbers mark the 1/12ths of an hour. But we are usually more interested in the 1/60ths of an hour: the minutes.

Since 12 and 60 differ by a factor of 5 (i.e. $12*5=60$), converting between the two is quite easy. We just multiply by 5.

Or as your daughter prefers, you could multiply by 10 (by adding a 0) and then divide by 2. Since $10/2=5$, it amounts to the same thing.

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